Bayesian Logistic Regression Models for Credit Scoring by Gregg Webster

 
Definition 3.6 
Let
 
( 
)
be a Markov chain and let 
. We then call 
-
the set 
Harris-recurrent if for all 
we have 
( 
| 
)
. 
-
the Markov chain Harris-recurrent if it is 
–irreducible for some probability distribution 
and whenever 
( )
, then 
is Harris-recurrent.
 
Lemma 3.4 
Let
 
( 
)
be a Markov chain with stationary distribution 
(with 
density 
). If 
and if the Markov chain is 
-irreducible and recurrent, then for any 
integrable function 
we have (with probability 1) 
∑ ( 
)
∫ ( ) ( )
( ( ))
for almost all starting values 
. If the Markov chain is Harris-recurrent, then the 
equation holds for all 
 
 
3.4.3 Markov chain Monte Carlo 
 
Markov chain Monte Carlo constructs a Markov chain that has as its stationary 
distribution, the target distribution. It does this by constructing an irreducible Markov 
chain, which ensures that most of the Markov chains resulting from an MCMC algorithm 
are recurrent or even Harris-recurrent. As explained, Harris recurrence ensures that the 
Markov chain converges to its stationary distribution for every starting value instead of 
almost every starting value. Thus, we need Harris recurrence to ensure that the MCMC 
algorithm converges. MCMC algorithms construct a transition kernel which results in a 

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Markov chain which is recurrent and converges to the target distribution. A general 
principle to do this is the Metropolis-Hastings (MH) algorithm. The Gibbs sampler is a 
special case of the MH algorithm.